It looks like their server is refusing your connection. What TLS version does your server use?
Create a test page with this code and post back with the result"
$ch = curl_init('https://tlstest.paypal.com/');
curl_setopt($ch, CURLOPT_SSL_VERIFYHOST, 2);
curl_setopt($ch, CURLOPT_USERAGENT, $_SERVER['HTTP_USER_AGENT']);
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, FALSE);
$data = curl_exec($ch);